If they are not real wood, what are they? Vinyl coated particleboard or laminate coated?
You might be better off to sand the existing paint, fill any bad spots, and paint over that.
If what you end up with is vinyl or laminate, sand it just enough to cut the gloss and prime with Zinsser's Cover Stain (oil). Then paint with oil enamel. Do the same if the existing finish is a latex.
For a lot more guidance, read the painting topic on my website at www.bobf.info. If you have questions after that, you can contact me through the website.